Errors in variables models in large noise regime




Date(s) : 19/09/2016   iCal
14 h 00 min - 15 h 00 min

The talk deals with estimating unknown parameters $a,b\in\mathbb{R}$ in the simple Errors-in-Variables (EiV) linear model $Y_i=a+bX_i+\epsilon_n\xi_i$; $ Z_i=X_i+\sigma_n\zeta_i$, where $i=1,\ldots,n$, $\xi_i,\, \zeta_i$ are i.i.d. standard Gaussian random variables, $X_i\in\mathbb{R}$ are unknown nuisance variables, and $\epsilon_n,\sigma_n$ are noise levels which are assumed to be known.
It is well known that the standard maximum likelihood estimates of $a,\, b$ haven’t bounded moments. In order to improve these estimates, we study the EiV model in the so-called Large Noise Regime assuming that $n\rightarrow \infty$, but $\epsilon_n^2=\sqrt{n}\epsilon_\circ^2$, $\sigma_n^2=\sqrt{n}\sigma_\circ^2$ with $\epsilon_\circ^2,\sigma_\circ^2>0$. Under these assumptions, the minimax approach to estimating $a,b$ is developed. In particular, it is shown that the minimax estimate of $b$ is a solution to a convex optimization problem and a fast algorithm for computing this estimate is proposed.

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